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1、相变晶体学张文征Lecture 1， part cO-lattice formulation and applications Dec. 4, 2014O-lattice and its application to understanding preferred interfacesBasic conceptUseful matrixes for calculationsO-lattice calculation g parallelism rules in reciprocal spaceExamplesOO- -lattice theory and applicationlattice
2、theory and applicationBollmann 1920 2009 瑞典物理学家 Inventor of O-latticeL= fcc 1 1 1L= bcc 1 1 0+=primary preferred state A 2D O-lattice formed by bcc It cross a single O-cell wallO-cell wall = a poor fit region, where dislocation possibly locateO-lattice point = zero misfit = center of good fit a regi
3、on, where a preferred state can sustain Use of O-lattice to calculate dislocationsPeriodic GMSC centers O-lattice elementsxiOprimary O-pointO-Cell wall2ndary O-pointsecondary CSL point Potential dislocationsPrimary OO- -lattice and lattice and primary dislocationsprimary dislocationsSecondary OO- -l
4、attice and lattice and secondary dislocationssecondary dislocationsFit (preferred state) vs Misfit (dislocations) Centers of fit = O-lattice (2D)Structure in GMSC vs preferred state (PS)Hypothesis: Periodicity in GSMC periodicity in PS, provided that in GMSC1. 2 periods basis vectors2.low CSL, 2 23.
5、GMSC/dense pl.Primary PS, 3D Secondary PS, 2D( = 15%)3D GMSC cell = unit cell of either lattice for I-PS, mainly for metal phases 2D GMSC cell carries the period of structural units only in interface for II-PS PS = fit GM pairs define lattice correspondence (CL) for fit Deviation between GM pairs sp
6、ecifies misfit GMSC principlex1 Oisotropic dilationConsistence of O-cell walls with dislocations in a 1D misfit systemO-cell wall traces =possible locations of dislocationsThe intersections of the O-cell walls with an interface are possible position of (misfit) dislocationsDB Williams and CB Carter1
7、996Square dislocation network of screw dislocations in a twist grain boundary in Si倾侧晶界 倾侧晶界纯扭转晶界Input needed for the O-lattice calculationBurgers vectors 柏氏矢量Misfit deformation field 错配场 in 3DAdditional Input needed for dislocation calculation金属界面的低能趋势：尽可能共格(成键), 包括共格/半共格/重位共格 /半重位共格/非共格 比喻：1D:在力所能
8、及的范围小朋友和大朋友尽可能手牵手取暖，连 续撑着的取暖团为择优态区,简称好区,连续好区之间边界称位错, 牵手对象之间的位移矢量= 柏氏矢量; 3D晶体中原子是多手观音, 择优态区的等价周期性结构的条件,限制了3D柏氏矢量的选择范围： 点阵平移矢量，或者DSCL矢量偏离择优态（无错配共格或者重位共格）的定量描述称错配场Interface orientation1.Lattice parameters (LP) 2.Orientation relationship (OR) between two phases 3.Preferred state in the interfacial struc
9、tureInput OR, LP, often assume primary preferred state (LC)Misfit depends only on lattice parameters when lattice base vectors are parallel 1-to-1 or a specified lattice correspondence is impliedThree elements for calculating misfit deformation filedMatrix Expressions of Matrix Expressions of Phase
10、TransformationPhase TransformationSome formulas and figures in web have been Some formulas and figures in web have been adopted, mainly from :adopted, mainly from : E E- -book and Lecture Notesbook and Lecture Notes H. K. D. H. Bhadeshia, University of CambridgeH. K. D. H. Bhadeshia, University of C
11、ambridgeHow to calculate interfacial misfit for How to calculate interfacial misfit for general cases?general cases?Vectors in direct and reciprocal spaceVectors in direct and reciprocal space Vectors in direct space are usually in Vectors in direct space are usually in form of column vectors in for
12、m of column vectors in Vectors in reciprocal space are in Vectors in reciprocal space are in form of row vectors ()form of row vectors ()(u1, u2, u3)Lattice correspondenceLattice correspondence Specifies fit vectors in the preferred stateSpecifies fit vectors in the preferred state The deviation bet
13、ween the corresponding vectors The deviation between the corresponding vectors defines the misfitdefines the misfit GMSC principleGMSC principle: a pair of points from different : a pair of points from different lattices are related by the llattices are related by the lattice correspondenceattice co
14、rrespondenceif they form a GMS within a GMSC exhibiting a if they form a GMS within a GMSC exhibiting a periodic structure surrounding the origin. periodic structure surrounding the origin. Bain correspondenceBain correspondenceBain correspondence: (3 none-linear related vectors) 1 0 0b|1 -1 0f/2, 0
15、 1 0b|1 1 0f/2, 1 1 1b/2|1 0 1f/2 Most fcc/bcc metallic phases are related by the Most fcc/bcc metallic phases are related by the Bain correspondence regardless the detailed ORBain correspondence regardless the detailed ORLattice correspondence for bcc/Lattice correspondence for bcc/hcphcp0 1 -1b |0
16、 0 0 1h 1 1 1b/2 |2 -1 -1 0h/3 -1 1 1b/2 |-1 2 -1 0h/3Lattice correspondenceLattice correspondenceLattice correspondence matrix: CLattice correspondence matrix: CThe corresponding vectors are related byThe corresponding vectors are related by u u = = C C u u (bold capital letters represent matrix) (bold capital letters represent matr
